Problem 29 - Calculus and its applications

Question

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int\left(x^{4}+4\right) e^{3 x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{3}(x^4 + 4)e^{3x} - \int v \, du + C \), where further integration by parts might be needed for full simplification.

Step 1: Choose a Method

The integral \( \int (x^4+4) e^{3x} \, dx \) can be approached using integration by parts. We'll use the formula \( \int u \, dv = uv - \int v \, du \). Here, we need to choose \( u \) and \( dv \) such that differentiating \( u \) simplifies the expression.

Step 2: Assign \( u \) and \( dv \)

Let \( u = x^4 + 4 \), which results in \( du = 4x^3 \, dx \). Choose \( dv = e^{3x} \, dx \) so that \( v = \frac{1}{3}e^{3x} \).

Step 3: Apply Integration by Parts

Using the integration by parts formula, \( \int u \, dv = uv - \int v \, du \), substitute \( u = x^4 + 4 \), \( du = 4x^3 \, dx \), \( dv = e^{3x} \, dx \), and \( v = \frac{1}{3}e^{3x} \).

Step 4: Calculate \( uv \) and \( \int v \, du \)

First, calculate \( uv = (x^4 + 4) \times \frac{1}{3}e^{3x} = \frac{1}{3}(x^4 + 4)e^{3x} \). Now find \( \int v \, du = \int \frac{1}{3}e^{3x} \times 4x^3 \, dx \).

Step 5: Evaluate \( \int \frac{4x^3}{3}e^{3x} \, dx \)

This step requires another integration by parts. Let \( u = x^3 \) and \( dv = \frac{4}{3}e^{3x} \, dx \). Then \( du = 3x^2 \, dx \) and \( v = \frac{4}{9}e^{3x} \). Apply the integration by parts formula again.

Step 6: Simplify the Evaluation

Once \( \int \frac{4x^3}{3}e^{3x} \, dx \) is evaluated through multiple integration by parts, substitute back into the expression for the overall integral to simplify. Carefully calculate these steps, keeping track of coefficients and constant multipliers. This may require several more iterations.

Step 7: Substitute Everything Together

Combine all parts from different iterations of integration by parts: \( \frac{1}{3}(x^4 + 4)e^{3x} - \text{Evaluated Part From Step 6} + C \). Simplify and finalize the solution.

Step 8: Differentiate to Check Your Work

Differentiate the solution to ensure the original integrand is obtained. This acts as a verification step to confirm that the integration was performed correctly.

Key Concepts

integration by parts

Integration by parts is a powerful technique used to integrate products of functions, especially when direct integration is complicated. The essence of integration by parts is derived from the product rule of differentiation. The formula for this technique is \[ \int u \, dv = uv - \int v \, du \]which allows us to transform the integral of a product into a generally easier form to evaluate.
To use integration by parts:

• Select functions for \(u\) and \(dv\) such that differentiating \(u\) and integrating \(dv\) simplifies the problem.
• Differentiate \(u\) to find \(du\) and integrate \(dv\) to find \(v\).
• Substitute into the integration by parts formula and solve the resulting integral.

In the given exercise, \(u = x^4 + 4\) and \(dv = e^{3x} \, dx\), where \(v\) becomes \(\frac{1}{3}e^{3x}\). Applying integration by parts here helps break down the tough integral into more manageable pieces.

substitution method in calculus

Although the primary solution uses integration by parts, the substitution method in calculus is another fundamental technique. It's designed to simplify integrals by making a substitution that transforms a given integral into a basic form.
Here's how you use substitution in calculus:

• Choose a substitution, \(u = g(x)\), where \(g(x)\) is part of the integrand.
• Find \(du\), which will replace \(dx\) in the integral, relating \(du\) to \(g'(x)\).
• Replace all instances of the original variable with \(u\) and its differential \(du\).
• Integrate with respect to \(u\), then substitute back to the original variable.

While our main task did not directly use a substitution, understanding this concept is essential, as it aids in handling scenarios where direct integration is challenging.

differentiation check

After solving an integral, it's crucial to verify the result. The differentiation check involves differentiating your solution, and checking if you recover the original integrand. This step is vital for ensuring no calculation mistakes were made.
To conduct a differentiation check:

• Differentiate the result of the integration, effectively applying the reverse process.
• Compare the differentiated expression to the original integrand.
• If they match, the integration process was likely correct; if not, reevaluate the calculations.

In this exercise, once the final result of the integration by parts is obtained, performing a differentiation check confirms that the original integrand \((x^4 + 4)e^{3x}\) is accurately recovered, validating the solution.